Displace modifier using an Image Texture on a Grid Mesh
In the modifier you should use Texture Coordinates > UV in order to control how the picture will print on the surface. So beforehand you must select your plane, go in Edit mode, select all and unwrap with U > Unwrap. I've also tested with a Subdivision Surface modifier above and below the Displacement modifier, it looks like it works better (note that your picture is very pixelated though so it will never be perfect).